lecture 21: chapter 9, section 1 inference for...
TRANSCRIPT
©2011 Brooks/Cole, CengageLearning
Elementary Statistics: Looking at the Big Picture 1
Lecture 21: Chapter 9, Section 1Inference for Categorical Variable:Confidence Intervals
3 Forms of InferenceProbability vs. ConfidenceConstructing Confidence IntervalSample Size; Level of Confidence
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.2
Looking Back: Review
4 Stages of Statistics Data Production (discussed in Lectures 1-4) Displaying and Summarizing (Lectures 5-12) Probability (discussed in Lectures 13-20) Statistical Inference
1 categorical 1 quantitative categorical and quantitative 2 categorical 2 quantitative
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.8
Four Processes of Statistics
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.9
Summarizing Categorical Sample Data (Review)
What proportion of sampled students ate breakfastthe day of the survey?
Looking Back: In Part 2, we summarized sample data forsingle variables or relationships.
Looking Ahead: In Part 4, our goal is to go beyond sampledata and draw conclusions about the larger populationfrom which the sample was obtained.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.10
Three Types of Inference Problem
In a sample of 446 students, 0.55 ate breakfast.1. What is our best guess for the population
proportion of students who eat breakfast?Point Estimate
2. What interval should contain the populationproportion of students who eat breakfast?Confidence Interval
3. Is the population proportion of students who eatbreakfast more than half (50%)?Hypothesis Test
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.11
Behavior of Sample Proportion (Review)For random sample of size n from population
with p in category of interest, sampleproportion has
mean p is unbiased estimator of p
(sample must be random)
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.13
Example: Checking if Estimator is Unbiased
Background: Survey produced sample proportion ofintro stat students (various ages and times of day) ata university who’d eaten breakfast.
Questions: Is the sample representative of all college students? All
students at that university? Were the values of the variable (breakfast or not) recorded
without bias? Responses:
Differences among college cafeterias, etc.______________________________________________
Question not sensitive __________________________
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.15
Example: Point Estimate for p
Background: In a representative sample ofstudents, 0.55 ate breakfast.
Question: What is our best guess for theproportion of all students at that universitywho eat breakfast?
Response: unbiased estimator for p_____ is best guess for p
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.17
Example: Point Estimate Inadequate Background: Our best guess for p, population
proportion eating breakfast, is sample proportion 0.55. Questions:
Are we pretty sure the population proportion is 0.55? By approximately what amount is our guess “off”? Are we pretty sure population proportion is > 0.50?
Responses: ___________________________________________ ______________________ ______________________
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.19
Beyond a Point Estimate
Sample proportion from unbiased sample isbest estimate for population proportion.Looking Ahead: For point estimate we don’tneed sample size or info about spread. Theseare required for confidence intervals andhypothesis tests, to quantify how good ourpoint estimate is.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.21
Probability vs. Confidence Probability: given population proportion,
how does sample proportion behave? Confidence: given sample proportion,
what is a range of plausible values forpopulation proportion?
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.23
Example: Probability Statement
Background: If students pick numbers from 1 to 20at random, p=0.05 should pick #7. For n=400, has mean 0.05 standard deviation shape approximately normal
Question: What does the “95” part of the 68-95-99.7Rule tell us about ?
Response: Probability is approximately 0.95 thatfalls within _________ of ______.
Looking Ahead: This statement about sample proportion is correct but notvery useful for practical purposes. In most real-life problems, we want to drawconclusions about an unknown population proportion.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.25
Example: How Far is One from the Other?
Background: An instructor can say abouthis/her position in the classroom:“I’m within 10 feet of this particular student.”
Question: What can be said about where thatstudent is in relation to the instructor?
Response:
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.26
DefinitionsMargin of Error: Distance around a sample
statistic, within which we have reason to believethe corresponding parameter falls.
A common margin of error is 2 s.d.s.Confidence Interval for parameter: Interval within
which we have reason to believe the parameterfalls = range of plausible values
A common confidence interval is sample statisticplus or minus 2 s.d.s.
A Closer Look: A parameter is not a R.V. It does not obey thelaws of probability so we must use the word “confidence”.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.28
Example: Confidence Interval for p
Background: 30/400=0.075 students picked #7“at random” from 1 to 20. Let’s assume sampleproportion for n=400 has s.d. 0.01.
Question: What can we claim about populationproportion p picking #7?
Response: We’re pretty sure p is__________________________________________
Looking Back: In Part I, we learned about biased samples. The data suggest p>0.05:students were apparently biased in favor of #7. Their selections were haphazard, notrandom. If sampling individuals or assigning them to experimental treatments is notrandomized, then we produce a confidence interval that is not centered at p.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.29
Level of Confidence Corresponds to Multiplier
By “pretty sure”, we mean “95% confident”,because 95% is the probability of sampleproportion within 2 s.d.s of p (for largeenough n).
Looking Back: Our probability statement claimed sampleproportion should fall within 2 s.d.s of population proportion.Now, the inference statement claims population proportionshould be within 2 s.d.s of sample proportion.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.30
Probability Interval for Picking #7
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.31
Confidence Interval for p Picking #7We do not sketch a curve showing probabilities for population proportionbecause it is not a random variable.
A Closer Look:How do we knowthe margin oferror?
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.32
Behavior of Sample Proportion (Review)For random sample of size n from population
with p in category of interest, sampleproportion has
mean p standard deviationWe do inference because p is unknown; how
can we know the standard deviation, whichinvolves p?
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.33
DefinitionStandard error: estimated standard
deviation of a sampling distribution.We estimate standard deviation of
with standard error .
Looking Ahead: In many situations throughoutinference, when needed information about thepopulation is unknown, we substitute knowninformation about the sample.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.34
Definition95% confidence interval for p: (approx.)
estimate= sample proportion
standard error
margin of error =2 standard errors
95% confidence intervalfor population proportion
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.35
Confidence Interval Formula: Conditions95% confidence interval for p: (approx.)
Sample must be unbiased(otherwise interval is not really centered at )
n must be large enough so is approx. normal(otherwise multiplier 2 from 68-95-99.7 Rule isincorrect)
Population size must be at least 10n(otherwise formula for s.d., which requiresindependence, is incorrect)
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.36
Conditions for Normality in Confidence Interval
Multiplier 2 from normal dist. approximatelycorrect if np and n(1-p) both at least 10.
But p is unknown so substitute :Require
Sample count in (X) and out (n-X) of categoryof interest should both be at least 10.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.38
Example: Checking Sample Size
Background: 30/400=0.075 students picked#7 “at random” from 1 to 20.
Question: Do the data satisfy requirement forapproximate normality of sample proportion?
Response:
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.40
Example: Checking Population Size
Background: To draw conclusions about criminalhistories of a city’s 750 bus drivers, a randomsample of 100 drivers was used.
Question: Is there approximate independence inspite of sampling without replacement, so formulafor standard error is accurate?
Response:
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.42
Example: Revisiting Original Question
Background: In sample of 446 college students,246 (proportion 0.55) ate breakfast.
Question: Assuming sample is representative, whatinterval should contain proportion of all students atthat university who eat breakfast?
Response: Approx. 95% confidence interval for p isLooking Back:Earlier we wondered ifa majority of studentseat breakfast. Theinterval suggests this isthe case, since it isentirely above 0.50.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.44
Example: Role of Sample Size Background: 95% confidence intervals based on
sample proportion 0.54 from various sample sizes:
Question: What happens as n increases? Response:
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.46
Example: A Common Margin of Error Background: Pollsters most often report a 3% error margin.
Question: What is the most common sample size for polls? Response: Approximately _______.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.47
Other Levels of ConfidenceConfidence level 95% uses multiplier 2. Other levels use
other multipliers, based on normal curve:
z
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.48
Other Levels of ConfidenceConfidence level 95% uses multiplier 2. Otherlevels use other multipliers, based on normal curve.More precise multiplier for 95% is 1.96 instead of 2.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.50
Example: Other Levels of Confidence Background: Of 108 students in committed
relationships, 0.70 said they took comfort by sniffingout-of-town partner’s clothing. Standard error canbe found to be 0.04.
Question: How do 90%, 95%, 98%, 99%confidence intervals compare?
Response: 90% C.I. is ___________________=(0.63, 0.77) 95% C.I. is ___________________=(0.62, 0.78) 98% C.I. is ___________________=(0.61, 0.79) 99% C.I. is ___________________=(0.60, 0.80)
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.52
Example: Other Levels of Confidence
Intervals get _____as confidence level increases:
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.53
Confidence Interval and Long-Run Behavior
Repeatedly set up 95% confidence interval forproportion of heads, based on 20 coinflips.
In the long run, 95% of the intervals shouldcontain population proportion of heads, 0.5.
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.54
Confidence Interval and Long-Run Behavior
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.56
Example: Confidence in the Long Run Background: “President-elect Barack Obama's campaign
strategists weren't the only ones vindicated Tuesday.Pollsters came out looking pretty good, too. Of 27 polls ofPennsylvania voters released in the campaign's final twoweeks, only seven missed Obama's 10.3-point victory bymore than their margins of error. Obama's national victory ofabout 6 points was within the error margins of 16 of the 21national polls released in the final week.”
Question: Should pollsters be pleased with success rates of20/27=16/21= 75% ?
Response:
pittsburghlive.com/x/pittsburghtrib/news/cityregion/s_597288.html
©2011 Brooks/Cole,Cengage Learning
Elementary Statistics: Looking at the Big Picture L21.57
Lecture Summary(Inference for Proportions: Confidence Interval)
3 forms of inference; focus on confidence interval Probability vs. confidence Constructing confidence interval
Margin of error based on standard error Conditions
Role of sample size Confidence at other levels Confidence interval in the long run